Financial Informatics

The following sections are included:

• Preface
• About the Editors
• Contents

Summary. A new approach to credit risk modelling is introduced that avoids the use of inaccessible stopping times. Default events are associated directly with the failure of obligors to make contractually agreed payments. Noisy information about impending cash flows is available to market participants. In this framework, the market filtration is modelled explicitly, and is assumed to be generated by one or more independent market information processes. Each such information process carries partial information about the values of the market factors that determine future cash flows. For each market factor, the rate at which true information is provided to market participants concerning the eventual value of the factor is a parameter of the model. Analytical expressions that can be readily used for simulation are presented for the price processes of defaultable bonds with stochastic recovery. Similar expressions can be formulated for other debt instruments, including multi-name products. An explicit formula is derived for the value of an option on a defaultable discount bond. It is shown that the value of such an option is an increasing function of the rate at which true information is provided about the terminal payoff of the bond. One notable feature of the framework is that it satisfies an overall dynamic consistency condition that makes it suitable as a basis for practical modelling situations where frequent recalibration may be necessary.

A new framework for asset price dynamics is introduced in which the concept of noisy information about future cash flows is used to derive the corresponding price processes. In this framework an asset is defined by its cash-flow structure. Each cash flow is modelled by a random variable that can be expressed as a function of a collection of independent random variables called market factors. With each such “X-factor” we associate a market information process, the values of which we assume are accessible to market participants. Each information process consists of a sum of two terms; one contains true information about the value of the associated market factor, and the other represents “noise”. The noise term is modelled by an independent Brownian bridge that spans the interval from the present to the time at which the value of the factor is revealed. The market filtration is assumed to be that generated by the aggregate of the independent information processes. The price of an asset is given by the expectation of the discounted cash flows in the riskneutral measure, conditional on the information provided by the market filtration. In the case where the cash flows are the dividend payments associated with equities, an explicit model is obtained for the share-price process. Dividend growth is taken into account by introducing appropriate structure on the market factors. The prices of options on dividend-paying assets are derived. Remarkably, the resulting formula for the price of a European-style call option is of the Black-Scholes-Merton type. We consider the case where the rate at which information is revealed to the market is constant, and the case where the information rate varies in time. Option pricing formulae are obtained for both cases. The information-based framework generates a natural explanation for the origin of stochastic volatility in financial markets, without the need for specifying on an ad hoc basis the dynamics of the volatility.

We consider a financial contract that delivers a single cash flow given by the terminal value of a cumulative gains process. The problem of modelling such an asset and associated derivatives is important, for example, in the determination of optimal insurance claims reserve policies, and in the pricing of reinsurance contracts. In the insurance setting, aggregate claims play the role of cumulative gains, and the terminal cash flow represents the totality of the claims payable for the given accounting period. A similar example arises when we consider the accumulation of losses in a credit portfolio, and value a contract that pays an amount equal to the totality of the losses over a given time interval. An expression for the value process of such an asset is derived as follows. We fix a probability space, together with a pricing measure, and model the terminal cash flow by a random variable; next, we model the cumulative gains process by the product of the terminal cash flow and an independent gamma bridge; finally, we take the filtration to be that generated by the cumulative gains process. An explicit expression for the value process is obtained by taking the discounted expectation of the future cash flow, conditional on the relevant market information. The price of an Arrow Debreu security on the cumulative gains process is determined, and is used to obtain a closed-form expression for the price of a European-style option on the value of the asset at the given intermediate time. The results obtained make use of remarkable properties of the gamma bridge process, and are applicable to a wide variety of financial products based on cumulative gains processes such as aggregate claims, credit portfolio losses, defined benefit pension schemes, emissions and rainfall.

An asymmetric information model is introduced for the situation in which there is a small agent who is more susceptible to the flow of information in the market than the general market participant, and who tries to implement strategies based on the additional information. In this model market participants have access to a stream of noisy information concerning the future return of an asset, whereas the informed trader has access to a further information source which is obscured by an additional noise that may be correlated with the market noise. The informed trader uses the extraneous information source to seek statistical arbitrage opportunities, while at the same time accommodating the additional risk. The amount of information available to the general market participant concerning the asset return is measured by the mutual information of the asset price and the associated cash flow. The worth of the additional information source is then measured in terms of the difference of mutual information between the general market participant and the informed trader. This difference is shown to be nonnegative when the signal-to-noise ratio of the information flow is known in advance. Explicit trading strategies leading to statistical arbitrage opportunities, taking advantage of the additional information, are constructed, illustrating how excess information can be translated into profit.

IN THIS ARTICLE, we present a method of generating interest rate dynamics from elementary economic considerations. There are of course numerous economic factors that affect the movement of interest rates, and causal relations that hold between these factors are often difficult to disentangle. So, rather than attempting to address a range of factors simultaneously, we will focus on one factor important in determining the interest rate term structure, namely the liquidity risk, in the narrow sense of cash demand. The interplay between liquidity and interest rates has long been discussed in economics literature (see, for example, Friedman, 1968). Our objective is to build an information-based model that reflects the market perception of future liquidity risk, and use it for the pricing and general risk management of interest rate derivatives…

We propose a model for the credit markets in which the random default times of bonds are assumed to be given as functions of one or more independent “market factors”. Market participants are assumed to have partial information about each of the market factors, represented by the values of a set of market factor information processes. Themarket filtration is taken to be generated jointly by the various information processes and by the default indicator processes of the various bonds. The value of a discount bond is obtained by taking the discounted expectation of the value of the default indicator function at the maturity of the bond, conditional on the information provided by the market filtration. Explicit expressions are derived for the bond price processes and the associated default hazard rates. The latter are not given a priori as part of the model but rather are deduced and shown to be functions of the values of the information processes. Thus the “perceived” hazard rates, based on the available information, determine bond prices, and as perceptions change so do the prices. In conclusion, explicit expressions are derived for options on discount bonds, the values of which also fluctuate in line with the vicissitudes of market sentiment.

The information–based asset–pricing framework of Brody–Hughston–Macrina (BHM) is extended to include a wider class of models for market information. To model the information flow, we introduce a class of processes called Lévy random bridges (LRBs), generalising the Brownian bridge and gamma bridge information processes of BHM. Given its terminal value at T, an LRB has the law of a Lévy bridge. We consider an asset that generates a cash–flow X_T at T. The information about X_T is modelled by an LRB with terminal value X_T . The price process of the asset is worked out, along with the prices of options. © 2010 Elsevier B.V. All rights reserved.

Abstract This paper presents an overview of information-based asset pricing. In the information-based approach, an asset is defined by its cash-flow structure. The market is assumed to have access to “partial” information about future cash flows. Each cash flow is determined by a collection of independent market factors called X-factors. The market filtration is generated by a set of information processes, each of which carries information about one of the X-factors, and eventually reveals the X-factor in a way that ensures that the associated cash flows have the correct measurability properties. In the models considered each information process has two terms, one of which contains a “signal” about the associated X-factor, and the other of which represents “market noise”. The existence of an established pricing kernel, adapted to the market filtration, is assumed. The price of an asset is given by the expectation of the discounted cash flows in the associated risk-neutral measure, conditional on the information provided by the market. When the market noise is modelled by a Brownian bridge, one is able to construct explicit formulae for asset prices, as well as semi-analytic expressions for the prices and greeks of options and derivatives. In particular, option price data can be used to determine the information flow-rate parameters implicit in the definitions of the information processes. One consequence of the modelling framework is a specific scheme of stochastic volatility and correlation processes. Instead of imposing a volatility and correlation model upon the dynamics of a set of assets, one is able to deduce the dynamics of the volatilities and correlations of the asset price movements from more primitive assumptions involving the associated cash flows. The paper concludes with an examination of situations involving asymmetric information. We present a simple model for informed traders and show how this can be used as a basis for so-called statistical arbitrage. Finally, we consider the problem of price formation in a heterogeneous market with multiple agents.

We consider a heat kernel approach for the development of stochastic pricing kernels. The kernels are constructed by positive propagators, which are driven by time-inhomogeneous Markov processes. We multiply such a propagator with a positive, time-dependent and decreasing weight function, and integrate the product over time. The result is a so-called weighted heat kernel that by construction is a supermartingale with respect to the filtration generated by the time-inhomogeneous Markov processes. As an application, we show how this framework naturally fits the information-based asset pricing framework where time-inhomogeneous Markov processes are utilized to model partial information about random economic factors. We present examples of pricing kernel models which lead to analytical formulae for bond prices along with explicit expressions for the associated interest rate and market price of risk. Furthermore, we also address the pricing of fixed-income derivatives within this framework.

When investors have heterogeneous attitudes towards risk, it is reasonable to assume that each investor has a pricing kernel, and that these individual pricing kernels are aggregated to form a market pricing kernel. The various investors are then buyers or sellers depending on how their individual pricing kernels compare with that of the market. In Brownian-based models, we can represent such heterogeneous attitudes by letting the market price of risk be a random variable, the distribution of which corresponds to the variability of attitude across the market. If the flow of market information is determined by the movements of prices, then neither the Brownian driver nor the market price of risk are directly visible: the filtration is generated by an ‘information process’ given by a combination of the two. We show that the market pricing kernel is then given by the harmonic mean of the individual pricing kernels associated with the various market participants. Remarkably, with an appropriate definition of Lévy information one draws the same conclusion in the case when asset prices can jump. As a consequence, we are led to a rather general scheme for the management of investments in heterogeneous markets subject to jump risk.

Lévy processes, which have stationary independent increments, are ideal for modelling the various types of noise that can arise in communication channels. If a Lévy process admits exponential moments, then there exists a parametric family of measure changes called Esscher transformations. If the parameter is replaced with an independent random variable, the true value of which represents a ‘message’, then under the transformed measure the original Lévy process takes on the character of an ‘information process’. In this paper we develop a theory of such Lévy information processes. The underlying Lévy process, which we call the fiducial process, represents the ‘noise type’. Each such noise type is capable of carrying a message of a certain specification. A number of examples are worked out in detail, including information processes of the Brownian, Poisson, gamma, variance gamma, negative binomial, inverse Gaussian and normal inverse Gaussian type. Although in general there is no additive decomposition of information into signal and noise, one is led nevertheless for each noise type to a well-defined scheme for signal detection and enhancement relevant to a variety of practical situations.

A heat kernel approach is proposed for the development of a novel method for asset pricing over a finite time horizon. We work in an incomplete market setting and assume the existence of a pricing kernel that determines the prices of financial instruments. The pricing kernel is modeled by a weighted heat kernel driven by a multivariate Markov process. The heat kernel is chosen so as to provide enough freedom to ensure that the resulting model can be calibrated to appropriate data, e.g. to the initial term structure of bond prices. A class of models is presented for which the prices of bonds, caplets, and swaptions can be computed in closed form. The dynamical equations for the price processes are derived, and explicit formulae are obtained for the short rate of interest, the risk premium, and for the stochastic volatility of prices. Several of the closed-form models presented are driven by combinations of Markovian jump processes with different probability laws. Such models provide a basis for consistent applications in various market sectors, including equity markets, fixed-income markets, commodity markets, and insurance. The flexible multidimensional and multivariate structure on which the resulting price models are based lends itself well to the modeling of dependence across asset classes. As an illustration, the impact of spiraling debt, a typical feature of a financial crisis, is modeled explicitly, and the contagion effects can be readily observed in the dynamics of the associated asset returns.

Numerous kinds of uncertainties may affect an economy, e.g. economic, political, and environmental ones. We model the aggregate impact by the uncertainties on an economy and its associated financial market by randomised mixtures of Lévy processes. We assume that market participants observe the randomised mixtures only through best estimates based on noisy market information. The concept of incomplete information introduces an element of stochastic filtering theory in constructing what we term “filtered Esscher martingales”. We make use of this family of martingales to develop pricing kernel models. Examples of bond price models are examined, and we show that the choice of the random mixture has a significant effect on the model dynamics and the types of movements observed in the associated yield curves. Parameter sensitivity is analysed and option price processes are derived. We extend the class of pricing kernel models by considering a weighted heat kernel approach, and develop models driven by mixtures of Markov processes.

We consider the filtering problem of estimating a hidden random variable X by noisy observations. The noisy observation process is constructed by a randomized Markov bridge (RMB) (Z_t)_{t∈[0, T]} of which terminal value is set to Z_T = X. That is, at the terminal time T, the noise of the bridge process vanishes and the hidden random variable X is revealed. We derive the explicit filtering formula, governing the dynamics of the conditional probability process, for a general RMB. It turns out that the conditional probability is given by a function of current time t, the current observation Z_t, the initial observation Z₀, and the a priori distribution ν of X at t = 0. As an example for an RMB, we explicitly construct the skew-normal randomized diffusion bridge and show how it can be utilized to extend well-known commodity pricing models and how one may propose novel stochastic price models for financial instruments linked to greenhouse gas emissions.

We model continuous-time information flows generated by a number of information sources that switch on and off at random times. By modulating a multi-dimensional Lévy random bridge over a random point field, our framework relates the discovery of relevant new information sources to jumps in conditional expectation martingales. In the canonical Brownian random bridge case, we show that the underlying measure-valued process follows jump-diffusion dynamics, where the jumps are governed by information switches. The dynamic representation gives rise to a set of stochastically-linked Brownian motions on random time intervals that capture evolving information states, as well as to a state-dependent stochastic volatility evolution with jumps. The nature of information flows usually exhibits complex behavior, however, we maintain analytic tractability by introducing what we term the effective and complementary information processes, which dynamically incorporate active and inactive information, respectively. As an application, we price a financial vanilla option, which we prove is expressed by a weighted sum of option values based on the possible state configurations at expiry. This result may be viewed as an information-based analogue of Merton’s option price, but where jump-diffusion arises endogenously. The proposed information flows also lend themselves to the quantification of asymmetric informational advantage among competitive agents, a feature we analyze by notions of information geometry.

In the information-based pricing framework of Brody, Hughston & Macrina, the market filtration {ℱ_t}_t≥0 is generated by an information process {ℱ_t}_t≥0 defined in such a way that at some fixed time T an ℱ_t-measurable random variable X_T is “revealed”. A cash flow H_t is taken to depend on the market factor X_t, and one considers the valuation of a financial asset that delivers H_t at time T. The value of the asset S_t at any time t ∊ [0, T) is the discounted conditional expectation of H_t with respect to ℱ_t, where the expectation is under the risk neutral measure and the interest rate is constant. Then S_T⁻ = H_t, and S_t = 0 for t ≥ T. In the general situation one has a countable number of cash flows, and each cash flow can depend on a vector of market factors, each associated with an information process. In the present work we introduce a new process, which we call the normalized variance-gamma bridge. We show that the normalized variance-gamma bridge and the associated gamma bridge are jointly Markovian. From these processes, together with the specification of a market factor X_t, we construct a so-called variance-gamma information process. The filtration is then taken to be generated by the information process together with the gamma bridge. We show that the resulting extended information process has the Markov property and hence can be used to develop pricing models for a variety of different financial assets, several examples of which are discussed in detail.

This paper introduces an information-based model for the pricing of storable commodities such as crude oil and natural gas. The model uses the concept of market information about future supply and demand as a basis for valuation. Physical ownership of a commodity is taken to provide a stream of convenience dividends equivalent to a continuous cash flow. The market filtration is assumed to be generated jointly by the following: (i) current and past levels of the dividend rate, and (ii) partial information concerning the future of the dividend flow. The price of a commodity is the expectation under a suitable pricing measure of the totality of the discounted risk-adjusted future convenience dividend, conditional on the information provided by the market filtration. In the situation where the dividend rate is modelled by an Ornstein—Uhlenbeck process, the prices of options on commodities can be derived in closed form. The approach that we present can be applied to other assets that yield potentially negative effective cash flows, such as real estate, factories, refineries, mines, and power generating plants.

Over the past decade, it has become evident that intentional disinformation in the political context—so-called fake news—is a danger to democracy. However, until now, there has been no clear understanding of how to define fake news, much less how to model it. This paper addresses both of these issues. A definition of fake news is given, and two approaches for the modelling of fake news and its impact in elections and referendums are introduced. The first approach, based on the idea of a representative voter, is shown to be suitable for obtaining a qualitative understanding of phenomena associated with fake news at a macroscopic level. The second approach, based on the idea of an election microstructure, describes the collective behaviour of the electorate by modelling the preferences of individual voters. It is shown through a simulation study that the mere knowledge that fake news may be in circulation goes a long way towards mitigating the impact of fake news.